Abstract
We consider a branching random walk on ℝ with a random environment in time (denoted by ξ). Let Z n be the counting measure of particles of generation n, and let \(\tilde Z_n (t)\) be its Laplace transform. We show the convergence of the free energy n −1log \(\tilde Z_n (t)\), large deviation principles, and central limit theorems for the sequence of measures {Z n }, and a necessary and sufficient condition for the existence of moments of the limit of the martingale \(\tilde Z_n (t)/\mathbb{E}[\tilde Z_n (t)|\xi ]\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Biggins J D. Martingale convergence in the branching random walk. J Appl Probab, 1977, 14: 25–37
Biggins J D. Chernoff’s theorem in the branching random walk. J Appl Probab, 1977, 14: 630–636
Biggins J D. The central limit theorem for the supercritical branching random walk, and related results. Stochastic Process Appl, 1990, 34: 255–274
Biggins J D, Kyprianou A E. Measure change in multitype branching. Adv Appl Probab, 2004, 36(2): 544–581
Chow Y, Teicher H. Probability Theory: Independence, Interchangeability, Martingales. Berlin: Springer-Verlag, 1995
Comets F, Popov S, On multidimensional branching random walks in random environments. Ann Probab, 2007, 35: 68–114
Comets F, Yoshida N. Branching random walks in space-time random environment: survival probability, global and local growth rates. J Theoret Probab, 2011, 24: 657–687
Dembo A, Zeitouni O. Large Deviations Techniques and Applications. New York: Springer, 1998
Guivarc’h Y, Liu Q. Propriétés asymptotiques des processus de branchement en environnement aléatoire. C R Acad Sci Paris, 2001, 332: 339–344
Hu Y, Shi Z. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann Probab, 2009, 37(2): 742–789
Huang C, Liu Q. Branching random walk with a random environment in time. Univ Bretagne-Sud, France, Preprint, 2013
Kaplan N, Asmussen S. Branching random walks I. Stochastic Process Appl, 1976, 4: 1–13
Kaplan N, Asmussen S. Branching random walks II. Stochastic Process Appl, 1976, 4: 15–31
Kuhlbusch D. On weighted branching processes in random environment. Stochastic Process Appl, 2004, 109: 113–144
Liang X, Liu Q. Mandelbrot’s cascades in random environments. Univ Bretagne-Sud, France, Preprint, 2013
Liu Q. Branching random walks in random environment. In: Ji L, Liu K, Yang L, Yau S -T, eds. Proceedings of the 4th International Congress of Chinese Mathematicians, 2007, Vol II. 2007, 702–719
Nakashima M. Almost sure central limit theorem for branching random walks in random environment. Ann Appl Probab, 2011, 21: 351–373
Shi Z. Branching random walks. Saint-Flour’s Summer Course, 2012
Yoshida N. Central limit theorem for random walk in random environment. Ann Appl Probab, 2008, 18: 1619–1635
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, C., Liang, X. & Liu, Q. Branching random walks with random environments in time. Front. Math. China 9, 835–842 (2014). https://doi.org/10.1007/s11464-014-0407-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-014-0407-1